# NAG Library Routine Document

## 1Purpose

e02bbf evaluates a cubic spline from its B-spline representation.

## 2Specification

Fortran Interface
 Subroutine e02bbf ( c, x, s,
 Integer, Intent (In) :: ncap7 Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: lamda(ncap7), c(ncap7), x Real (Kind=nag_wp), Intent (Out) :: s
C Header Interface
#include <nagmk26.h>
 void e02bbf_ (const Integer *ncap7, const double lamda[], const double c[], const double *x, double *s, Integer *ifail)

## 3Description

e02bbf evaluates the cubic spline $s\left(x\right)$ at a prescribed argument $x$ from its augmented knot set ${\lambda }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n+7$, (see e02baf) and from the coefficients ${c}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,q$in its B-spline representation
 $sx=∑i=1qciNix.$
Here $q=\stackrel{-}{n}+3$, where $\stackrel{-}{n}$ is the number of intervals of the spline, and ${N}_{i}\left(x\right)$ denotes the normalized B-spline of degree $3$ defined upon the knots ${\lambda }_{i},{\lambda }_{i+1},\dots ,{\lambda }_{i+4}$. The prescribed argument $x$ must satisfy ${\lambda }_{4}\le x\le {\lambda }_{\stackrel{-}{n}+4}$.
It is assumed that ${\lambda }_{\mathit{j}}\ge {\lambda }_{\mathit{j}-1}$, for $\mathit{j}=2,3,\dots ,\stackrel{-}{n}+7$, and ${\lambda }_{\stackrel{-}{n}+4}>{\lambda }_{4}$.
If $x$ is a point at which $4$ knots coincide, $s\left(x\right)$ is discontinuous at $x$; in this case, s contains the value defined as $x$ is approached from the right.
The method employed is that of evaluation by taking convex combinations due to de Boor (1972). For further details of the algorithm and its use see Cox (1972) and Cox and Hayes (1973).
It is expected that a common use of e02bbf will be the evaluation of the cubic spline approximations produced by e02baf. A generalization of e02bbf which also forms the derivative of $s\left(x\right)$ is e02bcf. e02bcf takes about $50%$ longer than e02bbf.

## 4References

Cox M G (1972) The numerical evaluation of B-splines J. Inst. Math. Appl. 10 134–149
Cox M G (1978) The numerical evaluation of a spline from its B-spline representation J. Inst. Math. Appl. 21 135–143
Cox M G and Hayes J G (1973) Curve fitting: a guide and suite of algorithms for the non-specialist user NPL Report NAC26 National Physical Laboratory
de Boor C (1972) On calculating with B-splines J. Approx. Theory 6 50–62

## 5Arguments

1:     $\mathbf{ncap7}$ – IntegerInput
On entry: $\stackrel{-}{n}+7$, where $\stackrel{-}{n}$ is the number of intervals (one greater than the number of interior knots, i.e., the knots strictly within the range ${\lambda }_{4}$ to ${\lambda }_{\stackrel{-}{n}+4}$) over which the spline is defined.
Constraint: ${\mathbf{ncap7}}\ge 8$.
2:     $\mathbf{lamda}\left({\mathbf{ncap7}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{lamda}}\left(\mathit{j}\right)$ must be set to the value of the $\mathit{j}$th member of the complete set of knots, ${\lambda }_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,\stackrel{-}{n}+7$.
Constraint: the ${\mathbf{lamda}}\left(j\right)$ must be in nondecreasing order with ${\mathbf{lamda}}\left({\mathbf{ncap7}}-3\right)>\phantom{\rule{0ex}{0ex}}{\mathbf{lamda}}\left(4\right)$.
3:     $\mathbf{c}\left({\mathbf{ncap7}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the coefficient ${c}_{\mathit{i}}$ of the B-spline ${N}_{\mathit{i}}\left(x\right)$, for $\mathit{i}=1,2,\dots ,\stackrel{-}{n}+3$. The remaining elements of the array are not referenced.
4:     $\mathbf{x}$ – Real (Kind=nag_wp)Input
On entry: the argument $x$ at which the cubic spline is to be evaluated.
Constraint: ${\mathbf{lamda}}\left(4\right)\le {\mathbf{x}}\le {\mathbf{lamda}}\left({\mathbf{ncap7}}-3\right)$.
5:     $\mathbf{s}$ – Real (Kind=nag_wp)Output
On exit: the value of the spline, $s\left(x\right)$.
6:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value  is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value  is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$, ${\mathbf{ncap7}}=〈\mathit{\text{value}}〉$ and ${\mathbf{lamda}}\left({\mathbf{ncap7}}-3\right)=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}\le {\mathbf{lamda}}\left({\mathbf{ncap7}}-3\right)$.
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$ and ${\mathbf{lamda}}\left(4\right)=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}\ge {\mathbf{lamda}}\left(4\right)$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{ncap7}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ncap7}}\ge 8$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

The computed value of $s\left(x\right)$ has negligible error in most practical situations. Specifically, this value has an absolute error bounded in modulus by , where ${c}_{\mathrm{max}}$ is the largest in modulus of ${c}_{j},{c}_{j+1},{c}_{j+2}$ and ${c}_{j+3}$, and $j$ is an integer such that ${\lambda }_{j+3}\le x\le {\lambda }_{j+4}$. If ${c}_{j},{c}_{j+1},{c}_{j+2}$ and ${c}_{j+3}$ are all of the same sign, then the computed value of $s\left(x\right)$ has a relative error not exceeding  in modulus. For further details see Cox (1978).

## 8Parallelism and Performance

e02bbf is not threaded in any implementation.

## 9Further Comments

The time taken is approximately ${\mathbf{c}}×\left(1+0.1×\mathrm{log}\left(\stackrel{-}{n}+7\right)\right)$ seconds, where c is a machine-dependent constant.
Note:  the routine does not test all the conditions on the knots given in the description of lamda in Section 5, since to do this would result in a computation time approximately linear in $\stackrel{-}{n}+7$ instead of $\mathrm{log}\left(\stackrel{-}{n}+7\right)$. All the conditions are tested in e02baf, however.

## 10Example

Evaluate at nine equally-spaced points in the interval $1.0\le x\le 9.0$ the cubic spline with (augmented) knots $1.0$, $1.0$, $1.0$, $1.0$, $3.0$, $6.0$, $8.0$, $9.0$, $9.0$, $9.0$, $9.0$ and normalized cubic B-spline coefficients $1.0$, $2.0$, $4.0$, $7.0$, $6.0$, $4.0$, $3.0$.
The example program is written in a general form that will enable a cubic spline with $\stackrel{-}{n}$ intervals, in its normalized cubic B-spline form, to be evaluated at $m$ equally-spaced points in the interval ${\mathbf{lamda}}\left(4\right)\le x\le {\mathbf{lamda}}\left(\stackrel{-}{n}+4\right)$. The program is self-starting in that any number of datasets may be supplied.

### 10.1Program Text

Program Text (e02bbfe.f90)

### 10.2Program Data

Program Data (e02bbfe.d)

### 10.3Program Results

Program Results (e02bbfe.r)